Dale A. Buralli, G. Michael Morris, and John R. Rogers
The optical properties of holographic kinoforms are described. It is shown that paraxial designs are notadequate for flNos. less than -F/10. A nonparaxial design is introduced that retains the high diffractionefficiency of the paraxial designs, yet also produces an unaberrated diffracted wavefront for the designwavelength. Aberration calculations and computer calculations, based on the Huygens-Fresnel principle, ofthe point spread functions for these elements show the necessity of using the nonparaxial design. Specifica-tions for a surface profile that takes account of the finite thickness of the diffracting surface are given. Amodel for kinoforms which can be used in optical design programs is proposed.
1. Introduction
The kinoform is a phase hologram in which thephase modulation is introduced by a surface reliefprofile.' In most kinoforms, the maximum reliefdepth is chosen such that, at the chosen design wave-length, the maximum phase modulation introduced bythe kinoform is 2r. Similar to other types of holo-grams, it is possible to use kinoforms as lenses.2 In thispaper we are concerned with rotationally symmetrickinoform lenses. Several precursors to the kinoformmay be found in the literature. These devices go byvarious names, including phase plates3 ,4 and phaseFresnel lenses.5 Even though these kinoform lensesare based on the well-known principles of Fresnelzones, we will not use the term Fresnel lens to describethese devices, since Fresnel lens, in the field of opticaldesign, usually refers to an optical element which uti-lizes incoherent superposition to form an image, whilekinoforms rely on coherent imagery. The readershould be forewarned however, that this choice of ter-minology is far from uniform in the literature concern-ing these elements. We review the design of kinoformlenses in the paraxial approximation and then extendthe design to the nonparaxial case. The optical pefor-mance of the two designs, as functions of both wave-length and f/No. are compared. Also, we show whatmodifications to the diffracting surface are required toaccount for the effects of the finite thickness of the
surface. We conclude with a suggestion concerningthe modeling of these devices when used with conven-tional optical design programs.
11. Paraxial Kinoform Design
As a first step in describing the kinoform opticalelement, we shall model the device as an infinitely thinphase screen. Since the kinoform is a pure phasehologram, the transmission function is a unit magni-tude, complex-exponential function. In the paraxialapproximation, a rotationally symmetric lens is de-scribed by a transmission function that has a quadraticdependence on the radial coordinate,6 i.e.,
tiens(xy;X) = exx[ -r(X+ (1)
In Eq. (1), x and y are the coordinates in the plane ofthe thin lens, X is the wavelength of the light, and f isthe focal length of the lens. We would like the kino-form to perform the same function as a lens, thus weexpect the kinoform to have a transmission functionsimilar to that in Eq. (1).
We recall that the Fresnel full-period zones are de-fined such that the optical path length from the edge ofthe mth zone is equal to f + mX0, where X0 is thewavelength for which the zones are defined. Thus, theequation that defines the locations of the zones in thex-y plane (r2 = x2 + y2 ) is
r2 = 2mXof + (mXO)2. (2)
In the paraxial region, r2 << (f/M)2, so Eq. (2) reduces to
rmparaxial = 2mX0f.When this work was done all authors were with University of
Rochester, Institute of Optics, Rochester, New York 14627; J. R.Rogers is now with Wild-Leitz, Heerbrugg, Switzerland.
We are therefore led to try a kinoform surface profilethat has a zone spacing dictated by Eq. (2') and asurface relief profile (blaze) that is parabolic withineach zone. The surface profile is obtained from thedesired phase transmission function by noting that
|O(xy) = (27r/Xo)OPD(x,y), where g(x,y) is the phaseof the transmission function and OPD is the opticalpath difference introduced by the element. Also,OPD = [n(Xo) - l]d(x,y), where n(Xo) is the index ofrefraction at X0 and d(xy) is the thickness of the ele-ment at the point (xy). Since the maximum phasemodulation introduced by the kinoform is to be 27r, themaximum OPD introduced is X0. Thus, the maximumsurface relief height dmax is
max n(X 0 )(3)
This is precisely the situation that has been analyzedby Dammann. 7 He shows that for a phase function,which we shall describe as
r2 \(r) = a27r m- 2 J) for rm < r < rm+l, (4)
a change of variables t = r2/(2Xof) transforms the phasefunction into
(Q) = a27r(m -t for m S ,¢ < m + 1. (5)
In Eqs. (4) and (5), the parameter a is the fraction of 27rphase delay that is introduced for wavelengths otherthan the design wavelength X0. More specifically,
Xo n() -1(6A n(o) -1 (6)
The important observation about Eq. (5) is that nowthe transmission function t( ) = exp[io( )] is a periodicfunction of t and can be expanded as a Fourier series.The result is
t(0) = exp[io()] = 3 cn exp(i2irnt), (7a)
n=-w
where
- exp-ir(a + n)] sin[ r(a + n)].lr(a +n)
(7b)
Since we are assuming an implicit time dependence ofexp(-iwt), a converging lens corresponds to a positivevalue off in Eq. (1). We would like positive values of nto correspond to converging diffracted orders so that apositive order represents a positive lens. Therefore,we change n to -n in Eqs. (7), reverse the order ofsummation, and make the reverse substitution for t.This gives
From Eqs. (8) and (1) we see that this element be-haves as a lens with an infinite number of focal lengths
fkinoform = 0 (10)x n
The efficiency in order n is given by
7 = Cnon = sinc(a 7 n). (11)
If a = 1, i.e., if X = Xo, then n = 1.0 for n = 1 and is zerofor all other orders. Thus, the element has a diffrac-tion efficiency of 100% for the design wavelength.Dammann also considers the case when the desiredparabolic profile is approximated by a series of discretesteps or sinusoids. Lenses using the discrete step ap-proximation have been fabricated by several laborato-ries.8 ' 9
Even though these elements are 100% efficient (forthe design wavelength), good optical performance islimited to relatively high f/Nos. owing to the paraxialdesign. Comparing the corresponding terms in thetransmission function and the phase terms in the bino-mial expansion of a spherical wave of radius o = of/Xn,i.e.,
(Psph(r) = A [- - ( 8 )+r X( ](12)
or using the aberration model for diffractive opticsdeveloped by Sweatt' 0 and Kleinhans,"1 we can calcu-late the amount of third-order wavefront sphericalaberration for the first diffracted order of the paraxialdesign kinoform operating at infinite conjugates:
= (4f/#)3 ) (13)
where h is the semiaperture of the element and f/# isthe f/number defined as the ratio of the focal length tothe clear aperture of the element. For third-order(primary) spherical aberration, the wavefront aberra-tion polynomial has the form W = 7r1(X)p 4 , where p isthe normalized radial pupil coordinate.
To determine the optical performance of these de-vices, a computer program was written to calculate thepoint spread function (PSF) of a kinoform. The cal-culation is performed using a numerical implementa-tion of the Huygens-Fresnel principle,'2 namely,
*'P1 (X,Y,z) = +- f J f '9(,nO) exp(ikR) X(O)dmd, (14)
where k = 2 7r/X, X(O) = cos(O) is the inclination factor, 0
is the angle between the refracted ray and the observa-tion point, and R is the propagation distance, R = [(x -()2 + (y - q)
2 + z2]1/2, in which z is the axial distancebetween the kinoform and the observation plane. InEq. (14), TI is the scalar field in the kinoform plane andTI is the field in the observation plane.
It is interesting to point out here why we chose toperform the numerical integration of Eq. (14) to calcu-late diffraction patterns, rather than make use of thecomputational efficiency of fast Fourier transform(FFT) routines. When solving diffraction problems,one is usually presented with integrals of the formgiven by Eq. (14). These integrals can result from theuse of the Huygens-Fresnel principle or a Rayleigh-Sommerfeld formula, based on the solution of theHelmholtz equation. In either case, we have a term ofthe form exp(ikR) in the integrand. The usual Fresnelapproximation is to expand R in a binomial series andretain only terms up to second order, i.e.,
Thus, in the Fresnel approximation, the diffractionintegral takes the form
IIIFCS(XYZ) =-i exikz + ik (x2 + y2)]
Xz
X FT{ I'(ts7n0)X() ex i . (2 + 72)]} X
(15b)
where FT indicates a Fourier transform and the fre-quency variables are given by f = x/(Xz) and f = /(Xz). In Eq. (15b) we have also made the approxima-tion 1/R 1/z. Equation (15b) is now in a form forwhich FFT techniques are suitable. However, the useof the Fresnel approximation has reduced the range ofvalidity of the calculation to the paraxial region. Weare interested in a more precise, nonparaxial calcula-tion. We can try to increase the accuracy of the calcu-lation and still retain the Fourier transform form of theintegral by writing.
exp(ikR) = exi i2W + yd)l exp + Iexp(ikR).
(16)
Then, Eq. (13) takes the form
qf'1(XYZ) = z FTJ4(I,17,0)X(0) exp[ikR + i2ir(fz + f011 -
(17)
We still have a Fourier transform, but the functionbeing transformed depends on the frequency vari-ables. For every set of output coordinates (xy), i.e.,for every frequency pair (ft,f0), the transform wouldhave to be recalculated. Since this eliminates thecomputational advantage of using FFT routines, wehave chosen a direct numerical implementation of Eq.(14) to calculate diffraction patterns.
PSFs for F/10 and F/5 paraxial-designed kinoforms,both with a focal length of 50 mm, are shown in Fig. 1.For these calculations the reference wavelength Xo ischosen to be the helium-d line (X0 = 0.58756 ptm). Thematerial for the kinoforms is PMMA (acrylic) plastic.The plots are calculated using an infinitely distant on-axis object, i.e., a normally incident plane wave. Theintensity of each PSF is normalized such that the peakintensity of the PSF for the reference wavelength andan unaberrated diffracted wave is 1.0. At F/10, we seean Airylike pattern for the PSF, with a slightly reducedpeak intensity due to a very small amount of sphericalaberration. However, at F/5, the effect of retainingonly the paraxial terms in designing the transmissionfunction is evident, and the PSF is far from diffrac-tion-limited. Even though this element is 100% effi-cient in diffracting all the incident light into one dif-fracted order, the resulting image in an F/5 element ishighly aberrated.
III. Nonparaxial Kinoform Design
Our goal is to find a surface profile that retains thehigh diffraction efficiency of the parabolic blaze andalso produces a high quality (unaberrated) diffractedwave. Our choice of profile is motivated by consider-
0.6
0.4
Third Order Spherical Aberration
nI= h ()j% 4 f#) (t)
F/10 it, = 0.0665 X
F/5 r, = 1.0637 x
0.21
0.005 0.010 0.015 0.020 0.025 0.030
Radius (mm)Fig. 1. Point spread functions for 50-mm focal length, paraxialdesign kinoforms. The solid line is an F/10 element; the dashed lineis an F/5 element. Each curve is normalized such that the peak
intensity for an unaberrated diffracted wavefront is 1.0.
ing the exact expression in the x-y plane for the OPD ofa converging spherical wave of radius f: OPD = (f2 +r
2)1/
2- f. Consider a phase function of the form
0(r) = a27r m- ) for r < r<rm+l, (18)
where the zone radius r is given in Eq. (2). Note thatEq. (4) is obtained by retaining only the first two termsof the binomial expansion of Eq. (18). If we make thechange of variables = [(f2 + r 2
)'/2 - ]/Xo, Eq. (18) has
the same form as Eq. (5). Now we have a 100% effi-cient element that produces a perfect spherical wavefor the design wavelength. The profile of Eq. (18) isexactly the profile obtained from a variational analysisof zone plates13 designed to maximize the intensity inthe first-order diffraction focus. From the above dis-cussion we see that the intensity is maximized becausewe have produced an unaberrated wavefront whichcontains all the incident energy. Performing a bino-mial expansion of the transmission function resultingfrom Eq. (18) {i.e., Eqs. (7) with = [(f2 + r 2
)1
/2
- ]/Xo1and comparing the appropriate terms in the expansionof a perfect spherical wave [i.e., Eq. (12)] allow us tocalculate the amount of third-order wavefront spheri-cal aberration in the first diffracted order for the phaseprofile of Eq. (18):
(4f/No.)3 ) (19)
Equation (19) shows that at X = Xo, 7r,(Xo) = 0. In fact,at the design wavelength, all orders of spherical aber-ration are zero. Again, the efficiency is given by i =sinc2(a - n). PSFs calculated using the surface profiledictated by Eq. (18) are shown in Fig. 2. One can seethat the PSF is essentially diffraction-limited for therange of apertures considered here.
It is worth pointing out that the resolution of themanufacturing process used to fabricate the kinoformdetermines the maximum feasible aperture. One caneasily show that the minimum zone width Smin is givenapproximately by8
In the visible, Xo - 0.5 ,um, so Smin f f/No. Am.Both the paraxial and nonparaxial kinoforms de-
scribed in this paper have been designed and evaluatedusing normally incident, plane wave illumination.Naturally, in optical systems, it may be required thatthe kinoforms operate at finite conjugate ratios. Thisshift of conjugates will change the amount of aberra-tion introduced by both types of kinoform. Also, thekinoform substrate material, acting as a plane-parallelplate, will introduce aberrations at finite conjugates.Both of these effects should be taken into account inthe design of any optical system which contains kino-form elements. Alternatively, if the conjugate ratio atwhich the kinoform is to operate is known, the zonespacing and blaze can be designed as to be correct forthe two conjugate points.
We now consider the wavelength dependence of kin-oforms. Again, there are two performance featuresthat are wavelength dependent: diffraction efficiencyand aberrations. The diffraction efficiency for ordern, for both paraxial and nonparaxial designs, is givenby Eq. (11). A plot of diffraction efficiency as a func-tion of wavelength for orders n = 0, 1, and 2 is given inFig.i 3. As pointed out earlier and as is evident fromthe figure, X = 1.0 for the first diffracted order whenused at the design wavelength X0.
The point spread functions for wavelengths otherthan the design wavelength will differ from the diffrac-tion limit because of lower efficiency and aberrations.An efficiency <100% will reduce the amount of energy
b)
Z,._n
CC:
F/1 0
7T1,F = -0.0210 X
7t1C =0.0165X
0.005 0.00 0.05 0.020 0.025 0.030
Radius (mm)
F/5
i = -0.3356 X
ictc = 0.2633 X
0.005 0.060RDleli- s/ ammo
0.015
Fig. 4. Point spread functions for nonparaxial design kinoforms for
three wavelengths. Each PSF was calculated in the paraxial focalplane for the wavelength of interest; (a) Ff10 kinoform; (b) Ff5
kinoform.
that is being focused to the focal plane for the diffract-ed order of interest, which is n = 1 in this case. Thevariation of spherical aberration with wavelength(spherochromatism) degrades the wavefront (to thirdorder) by the amount given by Eq. (19). The PSFs forthe helium-d, and hydrogen-F and C lines (XF =0.48613 ,um; Xc = 0.65627 Am) for F/10 and F/5 systemsare shown in Fig. 4. The point spread function foreach wavelength was calculated in the paraxial focalplane for that wavelength. Obviously, in any broad-
Fig. 5. Geometry and notation for the design of a finite thicknesskinoform.
Thus we see that in each zone the proper surface is ahyperboloid of revolution, with eccentricity
a2+ b2
e = \1 = n(Xo). (24)a
We can recast Eq. (22) into a form similar to thestandard optical design sag formula:
______ cr2=n(0) 1 1-(K +)c 2 r2 (25)
In Eq. (25), c and K are constants that define theparaxial curvature and conic constant, respectively.Explicitly, these parameters are given by
band system, the large amount of chromatic aberrationmust be corrected in the overall system design, since asingle kinoform cannot be achromatized all by itself.
IV. Finite Thickness Kinoform Design
Thus far, we have ignored the fact that the diffract-ing surface of the kinoform has some finite thickness,which will affect the phase modulation introduced bythe device. It is possible to take account of this finitethickness and the refraction of the light at the curvedzone surface in the specification of the surface profile.Instead of working from the desired phase modulation,we use Fermat's principle to design the blaze such thatthe optical path length from any point in a given zoneto the focal point is the same. Obviously, from zone tozone, the target optical path length will increase by onedesign wavelength, Xo. This process is similar to thedesign of a refracting surface which equates opticalpath lengths to produce a stigmatic imaging system fora given pair of conjugate points. The calculation is setup as in Fig. 5. The sagitta of the surface is to becalculated so that the optical path length in the mthzone is the same from any incident position in thatzone. The zone boundaries are given by the exactexpression of Eq. (2). To equate the optical pathlengths we set
-n(X0)s(r) + (f + mX0) = [f - s(r)]2 + r2. (21)
We want to solve Eq. (21) for s(r), the sagitta (or sag) ofthe diffracting surface. Note that s(r) is the directeddistance from the tangent plane to the surface; thisaccounts for the negative sign of the first term on theleft-hand side of Eq. (21). Equation (21) can be ma-nipulated into the form
[s(r) - so]2 r2
a - =1, (22)
where
n(Xo)[f + mX0 ] -fso=
n2() 0)- 1
1n2(Xo) - 1]2
2 [n(Xo)f - f -MXO]2
n 2(X) - 1
(23a)
(23b)
(23c)
1c= +
f[l -n(X,)] + mX0
K =-n2(Xo).
(26a)
(26b)
The first term on the right-hand side of Eq. (25) is justthe necessary correction so that the sag will go to zeroat the edge of each zone. [Recall that s(r) defines thesurface sag relative to the tangent plane.] If we solvefor the surface profile dictated by the thin, nonparaxialdesign, Eq. (18), by using 0(r) = (27r/Xo)[n(Xo) -l]Sthin(r), we find that sthin(r) has the same functionalform as Eq. (25), but the parameters c and K are givenas
- f[l - n(Xo)]
Kthin = -[n(X) - 1]2 - 1.
(27a)
(27b)
Comparing Eqs. (26) and (27), we see that takingaccount of the finite thickness of the element has dic-tated only slight modifications to the surface profile.The exact profile has a different paraxial curvature ineach zone and a slightly different conic constant that isunchanging from zone to zone. It is well known14 thatthe refracting surface, which will in the limit of geo-metrical optics form a perfect image of an infinitelydistant on-axis point, is a hyperboloid of revolutionwith c = 1/lf[1-n(o)]} and K =-n
2 (XO). Thus, we seethat the profile of Eqs. (25) and (26) is a combination ofthese perfect imaging hyperboloids, with different par-axial curvatures (and hence, different paraxial focallengths) to account for the removal of the excess 27rphase modulations. These surfaces can be thought ofas shells which intersect the optical axis at differentpoints. Practically speaking, the differences betweenthe profiles of Eqs. (26) and (27) are very slight, andonly become significant at f/Nos. at which the preci-sion of fabrication probably becomes the limiting fac-tor.
V. Modeling of Kinoforms in Optical Design Software
Conventional optical design software programs arebased on the tracing of exact rays through an opticalsystem. The complex surface of a kinoform makes raytracing more difficult than for a conventional, continu-ous optical surface. Also, since a typical kinoformconsists of hundreds of diffracting zones, many morerays need to be traced than for conventional optics.
Another potential problem is the specification of thesurface of the kinoform in a manner that the designprogram can recognize. Our goal is to develop a modelfor the imaging properties of a kinoform which is easilyimplemented into existing design programs.
Sweatt' 0 has shown that, for any diffracted order,the first-order imaging properties and the third-orderaberrations of a holographic lens are correctly modeledby an ultrahigh index lens. The equivalent model lensfor an optically recorded HOE has surface specifica-tions such that the element is corrected for sphericalaberration for the conjugates at which the hologramwas fabricated, with the additional requirement thatthe bending of the lens is determined by the substratecurvature. These model lenses have the property thatall rays from the object source point travel through themodel lens in a direction which is perpendicular to theHOE substrate. Since all the kinoforms described inthis paper are planar elements-designed for use withan infinitely distant object, the equivalent ultrahighindex lens is plano-convex, with the curvature of theconvex side determined such that the focal length isthe desired value. If the index of the equivalent lensfor the design wavelength X0 is n,(Xo), the proper curva-ture is
1 f~~l-ns(X0)] ~~~(28)f[1 - nlaXo)I
As has been shown in the previous sections, the parax-ial and nonparaxial design kinoforms have identicalfirst-order properties, but different aberration proper-ties. The choice of model lens surface curvature givenby Eq. (28) determines the paraxial focal length andthe aberration contributions associated with a spheri-cal surface. We must specify the asphericity, if any, ofthis surface such that the total aberration introducedby the equivalent lens is the same as that of the kino-form. Thus, the aspheric deformation of this surfaceof the equivalent lens is determined from the specifica-tion of the zone locations,"1 as we have seen that thechoice of paraxial or exact zone spacing determines theamount of geometrical aberration introduced by thekinoform. Generally, optical design programs requirethe description of a rotationally symmetric asphericsurface in terms of the conic constant (which is equal tothe negative of the square of the eccentricity), if theasphere is a conic section, or in terms of the coefficientsof a polynomial expression for the difference in sagittabetween the asphere and a base sphere for a generalrotationally symmetric asphere.
Comparing the aberration coefficients derived fromthe appropriate kinoform transmission functions [Eq.(13) for the paraxial design; Eq. (19) for the nonparax-ial design] with those of an ultrahigh index lens, we canfind the necessary aspheric deformation terms for thelens models of the two types of kinoform. (See anybook on geometrical optics for a discussion of the aber-rations of thin lenses and aspheric surfaces; for exam-ple, Ref. 15.) To find the appropriate coefficients, wecompare terms in the expansions of the phase func-tions of the paraxial and nonparaxial design kinoforms
with the aberration coefficients of a thin lens with one(possibly) aspheric surface when the index of refrac-tion approaches infinity. The equality of the aberra-tion coefficients requires that for a paraxial design, allaspheric coefficients of the convex surface are zero,whereas for the nonparaxial design, the conic constantis K = -n,2 (Xo). To account for the chromatic varia-tion of focal length, indicated by Eq. (10), the refrac-tive index of the ultrahigh index material, n(X), ismade linearly proportional to wavelength. Finally, wenote that in all realistic planar kinoforms the surfacerelief profile will be on one side of a plane-parallelsubstrate. One can easily show that a plane-parallelplate will introduce aberrations if used at finite conju-gates.16 To account for this, the ultrahigh index lensshould be placed in contact with a plane-parallel slabof the material of which the kinoform is made, with athickness equal to the kinoform substrate thickness.The refractive-index properties of this plate will bethose of the kinoform material, not the ultrahigh indexlens model material.
The model described above will correctly predict thelocation and aberrations of a kinoform-produced im-age, but not the energy distribution among the variousdiffracted orders. Also, the multiplicity of imagesresulting from the different orders has been ignored,since a refractive lens produces only one image. If it isdesired to calculate point spread functions, or otherdiffraction-based image parameters, the diffraction ef-ficiency of the kinoform must be considered. Ignoringthe effects of aberrations by using the Fresnel approxi-mation to the Huygens-Fresnel integral, i.e., Eq. (15),we can predict the form of a perfect point spreadfunction. The relative values of peak diffracted inten-sity, for an on-axis observation point, for each wave-length calculated from the Fresnel theory can be usedas spectral weighting factors for the wavelengths usedin the design program, so that the relative peak diffrac-tion intensities computed by the design programmatch those of the Huygens-Fresnel theory. Thisweighting reflects the diffraction efficiency in the or-der of interest but ignores the background illumina-tion resulting from the other diffracted orders. Thus,the diffraction patterns calculated by this model in-clude only the energy diffracted into the order of inter-est. If the amount of background illumination, i.e.,light diffracted into other orders, is large, this modelwill become more inaccurate. This will occur forwavelengths further removed from the design wave-length, i.e., for values of a significantly different fromunity [see Eq. (6)]. (It should be noted that the pre-dictions of scalar diffraction theory become increas-ingly inaccurate at extremely high apertures,'7 but thedifferences between the Huygens-Fresnel calculationand a more exact vector calculation are negligible forrelative apertures such that the f/No. is greater than-F/0.9, which includes all the elements considered inthis paper.) In the Fresnel approximation, we cancalculate explicity the on-axis intensity produced by akinoform for a normally incident plane wave. This isdone by using the transmission function resulting from
tion efficiency. The efficiency can be calculated fromEq. (11) for the wavelengths of interest.
As examples of the accuracy of this model, resultsfrom the Huygens-Fresnel calculation were comparedto point spread functions calculated, using the model,by ACCOS V,18 a commercial lens design program. InFig. 6, we see that the different procedures producevery similar results. We would expect the model togive erroneous results for wavelengths where the pa-rameter a is sufficiently different from one. If this isthe case, the background intensity due to the otherorders could not be ignored.
0.005 0.010Radius (mm)
b) 1.000
0.800
E
0.600
0.400
0.015
0.2001-_
0.005 0.010 0.015Radius (mm)
Fig. 6. Comparison of Huygens-Fresnel and ultrahigh index modelcalculations of point spread functions. The solid line is the Huy-gens-Fresnel result and the dashed line is the ACCOS v calculation;(a) nonparaxial design, F/5 kinoform, X = 0.48613 um (hydrogen-F
line); (b) paraxial design, F/5 kinoform, X = o = 0.58756 m.
Eq. (4) in the Fresnel diffraction integral, Eq. (15),with x = y = 0 to find the on-axis field produced by onediffracting zone. The result is then summed over allthe zones to find the total on-axis diffracted field. Theon-axis intensity I(0,0,z), normalized at the first-orderdiffraction focal point for I(O,0,f) = 1.0 when = , issin2( )_X ) ff_I(0,0,z) = J cos(ar)- sin(ar) 2 . (29)
(rN)2(1 - aXZ)2 tan[(rXdo)/(Xz)]J
In Eq. (29), N is the number of zones in the kinoform.At a focus, Zfocus = (Xof)/(Xn) (f is the design focallength for X = X0, and n is the diffracted order), and Eq.(29) reduces to
I(0,0, - ) = n2 sinc 2(a - n) = n2n. (30)
Equation (30) indicates that we should spectrallyweight the different wavelengths by an amount pro-portional to the diffraction efficiency times the squareof the order number n. In most cases, we will beconcerned with the first diffracted order (n = 1), so thewavelengths will be weighted by their relative diffrac-
VI. Summary
Kinoforms can serve as high efficiency, diffractiveoptical elements and can produce high quality wave-fronts. It is necessary to use a nonparaxial design forgood results when working at f/Nos. less than -F/10.The wavelength dependence of both diffraction effi-ciency and aberrations must be considered when kino-forms are to be used in an optical system. It is possibleto model kinoforms as ultrahigh index lenses in designprograms, so optical systems containing these ele-ments can be evaluated.
The authors gratefully acknowledge the support ofthis research by DARPA, the MIT/Lincoln Laborato-ry, and 3M Company. D. A. Buralli acknowledges thesupport of the Kodak Fellows Program.
Portions of this work were presented as paper 883-07at the SPIE O-E/LASE '88 Conference, Los Angeles,CA, 10-17 Jan. 1988.
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8. L. d'Auria, J. P. Huignard, A. M. Roy, and E. Spitz, "Photolitho-graphic Fabrication of Thin Film Lenses," Opt. Commun. 5,232(1972).
9. G. J. Swanson and W. B. Veldkamp, "Binary Lenses for Use at10.6 Micrometers," Opt. Eng. 24, 791 (1985).
10. W. C. Sweatt, "Describing Holographic Optical Elements asLenses," J. Opt. Soc. Am. 67, 803 (1977).
11. W. A. Kleinhans, "Aberrations of Curved Zone Plates and Fres-nel Lenses," Appl. Opt. 16, 1701 (1977).
12. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1980), pp. 370-375.
13. R. Tatchyn, P. Csonka, and I. Lindau, "A Unified Approach tothe Theory and Design of Optimum Transmission DiffractionSystems in the Soft X-Ray Range," Proc. Soc. Photo-Opt. In-strum. Eng. 503, 168 (1984).
14. R. Kingslake, Lens Design Fundamentals (Academic, Orlando,1978), pp. 112-113.
15. W. T. Welford, Aberrations of Optical Systems (Adam Hilger,Bristol, 1986), pp. 152-153 and 226-234.
16. Ref. 15, pp. 234-235.17. See, for example, H. H. Hopkins, "The Airy Disk Formula for
Systems of High Relative Aperture," Proc. Phys. Soc. London55, 116 (1943); M. Mansuripur, "Distribution of Light at andNear the Focus of High-Numerical-Aperture Objectives," J.Opt. Soc. Am. A 3, 2086 (1986).
18. ACCOS V is a trademark of Scientific Calculations, Inc., 7635
Main St., Fishers, New York 14453.
Meetings continued from page 864
1989August
25-29 Int. Conf. on Solid Surfaces. Cologne A. Benningho-ven, Physikalisches Institut der Universitat Mun-ster, Wilhem-Klemm-Strasse 10, D-4000 Munster,F.R. Germany
20-25 9th Int. Conf. on Crystal Growth, Sendai Secretariat:9th Int. Conf. on Crystal Growth, co Inter GroupCorp., 8-5-32 Akasaka, Minato-ku, Tokyo 107, Japan
22-26 10th Int. Symp. on Nuclear Quadrupole ResonanceSpectroscopy, Takayama T. Asaji, Dept. of Chem.,PC 11, Faculty of Science, Nagoya U., Chikusa, Na-goya 464-01, Japan
26-31 7th Int. Summer School on Crystal Growth, Zao H.Komatsu, Inter Group Corp., Akasaka YamakatsuBldg., 8-5-32 Akasaka, Minato-ku, Tokyo 107, Japan
27-31 3rd Int. Symp. on Foundation of Quantum Mechanics-In the Light of New Tech., Tokyo H. Ezawa, Dept. ofPhysics, Gakushuin U., Mejiro, Toshima-ku, Tokyo171, Japan
28-31 7th Int. Conf. on Dynamical Processes in Excited Statesof Solids, Athens, GA J. Rives, Physics & AstronomyDept., U. of GA, Athens, GA 30602
25-29 16th Int. Symp. on Gallium Arsenide & Related Com-pounds, Karuizawa T. Katoda, Res. Center for Ad-vanced Science & Tech., U. of Tokyo, 4-6-1 Komaba,Meguro-ku, Tokyo 153, Japan
26-28 Int. Symp. on Optical Memory, Kobe Secretariat, coBusiness Center for Academic Societies Japan, 3-23-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
26-29 3rd Int. Conf. on Laser Anemometry Advances & Appli-cations, Wales L A Conf. 1989, Dept. of Engineering,U. of Manchester, Manchester M13 9PL, England
7-10 7th Int. Congr. Applications of Lasers & Electrooptics,Boston Laser Inst. Am., 5151 Monroe St., Toledo,OH 43623
15-20 Optics '89: OSA Ann. Mtg., Orlando OSA Mtgs.
September Dept., 1816 Jefferson Pl., NW, Wash., DC 20036
4-8 ISES Solar World Congress, Kobe Secretariat, Int.Communications, Inc., Kasho Bldg., 2-14-9 Nihonba-shi, Chuo-ku, Tokyo 103, Japan
5-8 O-E/Fibers '89 Symp. Optoelectronics & Fiber OpticDevices & Applications, Boston SPIE, P.O. Box 10,Bellingham, WA 98227
9-14 2nd Int. Symp. on Rare Earth Spectroscopy, ChangchunS. Qiang, Changchun Inst. of Applied Chem., Acade-mia Sinica, Changchun 130022, China
11-15 European Conf. on Optical Communication, Gothen-berg ECOC '89, Chalmers U. of Tech., S-41296 Goth-enberg, Sweden
14-16 Int. Symp. on Noise & Clutter Rejection in Radar &Imaging Sensors, Kyoto T. Suzuki, Dept. of Elec-tronics, U. of Electro-Communications, Chofu-shi,Tokyo 182, Japan
23-29 Advanced Processing Tehnologies for Optical & Elec-tronic Devices Mtg., Santa Clara SPIE, P.O. Box 10,Bellingham, WA 98227
18-21 Pacific Conf. on Chemistry & Spectroscopy, PasadenaW. Carter, P.O. Box 5732, Pasadena, CA 91107
22-26 Int. Conf. on Semiconductor & Integrated Circuit Tech.,Beijing Cont. Ed. in Eng., U. Ext., U. of CA, 2223Fulton St., Berkeley, CA 94720
November
5-10 1989 Advances in Intelligent Robotics Systems, Phila-delphia SPIE, P.O. Box 10, Bellingham, WA 98227
7-10 36th AVS Natl. Vacuum Symp., Phoenix AVS, 335 E.45th St., New York, NY 10017
9-11 Optical Storage for Small Systems Mtg., Los AngelesTOC, P.O. Box 14817, San Francisco, CA 94114
12-17 5th Int. Congr. on Advances in Non-Impact PrintingTechnologies, San Diego SPSE, 7003 Kilworth La.,Springfield, VA 22151
